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1. Bitcoin Spreads Like a Virus March 20, 2019 Timothy F. Peterson, CFA, CAIA Cane Island Alternative Advisors, LLC 515A South Fry Rd, #191 Katy, TX 77450 +1 (832) 5443529 timothy@caneisland.com About the author. Timothy is the founder and investment manager of Cane Island Alternative Advisors, which manages cryptocurrency and global macro strategies. His experience includes serving as CFO for a Houstonbased financial firm where he had responsibility for comanaging an optionhedged equity strategy. Timothy is an emerging expert on cryptocurrency investment and valuation and has authored the very popular "Metcalfe's Law as a Model for Bitcoin's Value," as well as a book “Performance Measurement for Alternative Investments” (Risk Books: 2015.) He currently serves as Chapter Executive for the Houston CAIA Chapter. Timothy is a Chartered Financial Analyst, Chartered Alternative Investment Analyst, and holds an M.S. Finance, and a B.A. Economics from the University of Colorado. Bitcoin Spreads Like a Virus March 20, 2019 Page 1 of 14 v1.2 Exposure Draft
2. Bitcoin Spreads Like a Virus March 20, 2019 Abstract. We illustrate, by way of example, that Bitcoin’s longterm price is nonrandom and can be modeled as a function of the logistic growth of number of users n over time. Using observed data for both Facebook and Bitcoin, we derive the relationships between price, number of users, and time, and show that the resulting market capitalizations likely follow a Gompertz sigmoid growth function. This function, historically used to describe the growth of biological organisms like bacteria, tumors, and viruses, likely has some application to network economics. We conclude that the longterm growth rate in users has considerable effect on the longterm price of bitcoin. Keywords: bitcoin, cryptocurrency, valuation, growth model, network economics, network effect. JEL Codes: C22, C49, C51, C53, C58, D85, E40, E42, E49, E51, G12, G17, L14 Bitcoin Spreads Like a Virus March 20, 2019 Page 2 of 14 v1.2 Exposure Draft
3. Noise trading is trading on noise as if it were information. People who trade on noise are willing to trade even though from an objective point of view they would be better off not trading. Perhaps they think the noise they are trading on is information. Or perhaps they just like to trade…[T]hey don't know they are trading on noise. They think they are trading on information. —Fischer Black, Noise If there were ever a noisy market, the young cryptocurrency environment fits the definition. Rarely do we witness such competing and sometimes concerted efforts to both establish and destroy a new asset class. The participants in this cacophony of change in the financial structure span the spectrum of naïve neophytes and phenomenal prodigies. At one end are young, arrogant daytrading chartists who seek to maximize their social media audience. At another are sophisticated, longestablished financial institutions with millennia of combined experience and quantitative computing power beyond belief. In between are governments, consumers, opportunists, celebrities, crooks, and investors, each weighing in on cryptocurrency with their varying degrees of experience, means, and motive. This paper offers a simple explanation of price formation in the burgeoning and oft misunderstood cryptocurrency ecosystem. Using bitcoin as an example, we provide convincing empirical evidence that price formation is not a semirandom result of emotional investing but instead is founded on economic principles of value that have only recently begun to be recognized: network economics. For over a decade, bitcoin’s price, as well as the viability and merit of the Bitcoin network itself, has been derided by classical economists as a fiction. Criticisms of bitcoin range from purely speculative investment to outright fraud. There is a common yet misinformed belief, even among ardent supporters and holders of bitcoin, that the true value of bitcoin is at best unknown and at worst not knowable or zero. No modern currency, not even sovereign fiat currency, has intrinsic or fundamental value. Every currency in widespread use today is representative money. Commodity money has other characteristics that impart value, namely scarcity and durability, that paper money does not. However, lack of intrinsic value is not the same as having no value. The Federal Reserve Bank of St. Louis [Andolfatto, 2019] explains the lack of fundamental value is a perfectly normal and acceptable characteristic of both representative and commodity money: Gold, for example, trades above its value as measured by its industrial applications. The U.S. dollar trades above its fundamental value in discharging U.S. tax obligations. The premium some people are willing to pay for gold and the U.S. dollar reflects the value these objects possess as exchange media. The market value of these objects would decline, but not fall to zero, should this premium suddenly vanish. An examination of bitcoin prices offers some interesting observations that directly counter the “value is a mystery” myth. The first is that, as proponents have long argued, the value of a currency is primarily dependent upon use and acceptance of that currency. Again, citing the Federal Reserve: What is the source of the fundamental value of Bitcoin? Think about it this way. At its core, Bitcoin is a database management system. Database management systems can have a fundamental value if they are tailored to meet the needs of a given constituency. Bitcoin offers people a money storage and transfer system with two key properties: (i) permissionless access and (ii) decentralized database management. The first property means that no one can prevent a user from sending any amount of Bitcoin from one account to another. The second property means that the protocol does not depend on the existence of a delegated authority to manage accounts and transfer funds. The Bitcoin Spreads Like a Virus March 20, 2019 Page 3 of 14 v1.2 Exposure Draft
4. fundamental demand for Bitcoin derives from the fact that there are at least some people who value these features. This fundamental demand provides a nonzero lower bound on the price of Bitcoin. This hypothesis has been tested and is also apparent from a cursory examination of the relationship between bitcoin’s price and activity associated with the Bitcoin payment network. Specifically, price changes tend to be highly correlated with changes in number of wallets, active addresses, unique addresses, and transaction activity. Using daily data since January 2012 – February 2019, these correlations are summarized in Table 1. Table 1 Correlations of Lognormal Returns in the Bitcoin Network 60day periods from January 2012February 2019 txn count active addresses unique addresses wallets transactions price (USD) 0.15 0.58 0.44 0.19 0.20 txn count active addresses unique addresses wallets 0.71 0.64 0.36 0.85 0.73 0.35 0.63 0.25 0.84 0.28 Sources: coinmetrics.io and blockchain.info By way of numerical example, we show that bitcoin’s longterm price is a function consistent with the logistic growth of number of users n over time. From observed data, we derived the relationships between price, number of users (proxied as active addresses), and time, and show that the resulting market capitalization is consistent with a Gompertz sigmoid growth function. The data were obtained entirely from coinmetrics.io and blockchain.info, and consist of daily prices and active accounts from July 2010 through February 2019. THE NETWORK EFFECT IN BITCOIN Metcalfe’s law is based on the mathematical tautology describing connectivity among n users. Hence, network value P is a function of number of users n. As more people join a network, they add to the value of the network nonlinearly; i.e., the value of the network is proportional to the square of the number of users. The underlying mathematics for Metcalfe’s law is based on pairwise connections (e.g., telephony). If there are four people with telephones in a network, there could be a total of 3 + 2 + 1 = 6 connections. This law, like most other laws, assumes equality among the members’ network connections. The law is commonly expressed in shorthand as n2, which is the approximate value of P when n is large. However, Metcalfe [2013] cautions that growth in P is subject to logistic decay, a function he labeled affinity (A). Alternatively, we could say that n grows nonlinearly consistent with a sigmoid function. Compared to most economic theories, Metcalfe’s law is relatively untested. Van Hove [2014] argued it was correct, while Zhang [2015], Peterson [2018], and Pele [2019] have all empirically confirmed Metcalfe’s law. Peterson and Pele both asserted the law is only useful over long measurement periods, an unsurprising qualification in light of Black [1986] quoted at the beginning of this article. Bitcoin Spreads Like a Virus March 20, 2019 Page 4 of 14 v1.2 Exposure Draft
5. The most commonly cited dispute of Metcalfe’s law is Briscoe [2006], who argue Metcalfe’s law as too optimistic and offer a subdued n log(n) model as an alternative. Their approach attempted to incorporate the concept of diminishing marginal return central to neoclassical economics. However, Briscoe [2006] does not derive the model and offers no empirical testing or results. Metcalfe [2013] counters that the diminishing incremental value is already captured in his A coefficient. Zhang [2015], Van Hove [2016] and Peterson [2018] concluded Metcalfe’s law was a more parsimonious and quantitatively satisfying model. Metcalfe’s law determines the value of a network for a given n. It says nothing about the growth rate of n. We consider three possibilities: If n grows linearly, then P ≈ n2. Metcalfe has said this assumption is unrealistic as P would grow without bound. However, in periods of early adoption, this model is probably a close approximation of value. P is bounded by logistic decay. n is subject to logistic growth. This approach is the subject of this paper. It is likely than n grows at a small rate, then a large rate, then a small rate, indicating stages of adoption as indicated in telephony by Chaddha [1971], Bewley, [1988], and Islam [2002]. Zhang [2015] used a similar function to model Facebook and Tencent market capitalizations. This bellshaped pattern of growth rates, on a cumulative basis, gives rise to a sigmoid function: 𝑃 𝑛 𝛼𝑒 1 To illustrate the differences, generic plots of three variations of Metcalfe’s law are show in Figure 1. The first is the simple approximation of Metcalfe’s law. The second is Metcalfe’s revision to his law by incorporation of a logistic decay factor A, and the third is a Gompertz function. Figure 1 Bitcoin Spreads Like a Virus March 20, 2019 Page 5 of 14 v1.2 Exposure Draft
6. A NUMERICAL EXAMPLE USING BITCOIN AND FACEBOOK We start by observing two relationships integral to the pricing of bitcoin. First, bitcoin’s lognormal price P with respect to time t is a nearperfect horizontal parabolic arc. A horizontal parabola is 𝑥 𝑎 𝑦 𝑘 ℎ 2 where (h, k) is the vertex. Plotting ln(Pt) and setting a = 1, h = 0, and k = 0 we have ln 𝑃 1 √𝑡 ln 𝑃 0 0 3 𝑡 4 which we have graphed in Figure 2. Figure 2 Bitcoin Price 100,000.00 10,000.00 1,000.00 100.00 10.00 1.00 0.10 0.01 Days since July 16, 2010 clean price manipulated price ln(p) = a(yk)^2 + h = t The elegance of this relationship between price and time is by no means a coincidence. It is rare that one has an opportunity to view the gradual adoption of a currency (or other asset) over time. This is in part because many companies are private during their development stages. But using wellknown Facebook (FB) we can see it follows a similar pattern (Figure 3). Bitcoin Spreads Like a Virus March 20, 2019 Page 6 of 14 v1.2 Exposure Draft
7. Figure 3 Facebook Price Facebook Price (P) 1,000 100 10 1 0 5 10 15 20 25 30 35 40 45 Quarters since Inception (t) Observed Facebook Price (FB) Parabolic Curve Figure 4 Facebook User Growth log(n) = log (sqrt(t)) 15 log (Monthly Active Users) 14 13 y = 3.571x + 8.042 R² = 0.9952 12 11 10 9 8 7 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 log scale square root time Source: https://www.statista.com/statistics/264810/numberofmonthlyactivefacebookusersworldwide/ Bitcoin Spreads Like a Virus March 20, 2019 Page 7 of 14 v1.2 Exposure Draft
8. The relationship in Figure 3 is a consequence of Metcalfe’s law. In Figure 4, we plot the Metcalfe value P of Facebook Monthly Active Users (n) where 𝑛 𝑛 1 5 2 As we will see later, the relationships that describe Bitcoin’s value follow the exact same mathematical growth principles as those of Facebook. Facebook Figure 3 corresponds to Bitcoin Figure 1. Facebook Figure 4 corresponds to Bitcoin Figure 7. Facebook pricing prior to the IPO is unavailable. The value can only be estimated with Metcalfe’s law, and Figure 4 lays the groundwork for that analysis. We do not analyze Facebook’s value in this paper. We only use it to show the persistence of the network effect outside of cryptocurrency. Referring back to Figure 2, there are three notable exceptions where bitcoin’s price deviated from the parabolic trend. These are periods of documented price manipulation and eventual resolution to equilibrium. These periods of price manipulation have been wellresearched and described in Gandal [2017], Griffin [2018], and Peterson [2018], as well as the financial press [Leising, 2018 and Robinson, 2018]. Using the method described in Peterson, we plot the deviation from equilibrium value to better illustrate the four periods of price manipulation, shown in Figure 5. Accordingly, we exclude these periods (identified by local price minima preceding and subsequent to drawdown) from our analysis. 𝑃 Figure 5 Bitcoin Spreads Like a Virus March 20, 2019 Page 8 of 14 v1.2 Exposure Draft
9. Using periods where Bitcoin’s price was in equilibrium, we can easily see ln 𝑃 𝑎√𝑡 𝑏 6 which we plot in Figure 6. Figure 6 ln(p) = sqrt(t) 60day "clean" periods 26.0 R² = 0.9945 24.0 Log BTC Market Cap 22.0 20.0 18.0 16.0 14.0 12.0  1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 square root time Growth in Bitcoin Users From Figure 7, we observe that n = f(t): ln √𝑡 𝑐 ln 𝑛 𝑑 7 This relationship can also be expressed as √𝑡 𝑒 𝑐 ln 𝑛 𝑑 8 Substituting Equation 8 into Equation 6, we have Bitcoin Spreads Like a Virus March 20, 2019 Page 9 of 14 v1.2 Exposure Draft
10. Figure 7 Bitcoin Adoption Rate Since 1 January 2011 15.0 R² = 0.979 14.0 Log (active accounts) 13.0 12.0 11.0 10.0 9.0 8.0 7.0 6.0 2.50 2.70 2.90 3.10 3.30 3.50 3.70 3.90 4.10 4.30 log scale square root time Clean Price Periods Manipulated Price Periods 𝑎𝑒 𝑐 ln 𝑛 ln 𝑃 𝑑 𝑏 ln(n) = ln(sqrt(t)) 9 which is in the form 𝑃 where 𝛾 𝛾𝑛 𝛼𝑒𝛽𝑒 1 %∆𝑛 10 As previously mentioned, Equation 1 is a Gompertz function, a logistic function that for decades has been used to model viral infection [Scott, 1959], bacterial growth [Murphy, 1973], tumor growth [Brunton, 1980], and mobile phone proliferation [Islam, 2002]. The most notable application of a Gompertz function to bitcoin pricing to date is Peterson [2018] who used it to model Metcalfe’s affinity coefficient. Bitcoin Spreads Like a Virus March 20, 2019 Page 10 of 14 v1.2 Exposure Draft
11. Relationship to Metcalfe’s Law Using coefficient notation from Equations 5 and 6, and using possible paired connections notated as n , the relationship between network value and network size can be expressed as 2 ln ln 𝑃 𝑏 𝑐′ln 𝑛 𝑑′ 𝜀 11 We plot this relationship in Figure 8. Analysis of Bitcoin Price and User Data Using a linear firstdifferences approach (60day intervals) to control for autocorrelation in the time series, we obtain estimates for c’ and d’ in Equation 11 and present the results in Table 2 and Figure 9. This regression is constrained such that the intercept d’ = 0, a condition we found likely to be true when running an unconstrained regression. Figure 8 Equation 11 log[log(p)] vs. log(n2) 3.00 R² = 0.9768 2.50 log(log(p)b) 2.00 1.50 1.00 0.50 10 12 14 16 18 log Bitcoin Spreads Like a Virus March 20, 2019 20 22 24 26 28 (n2) Page 11 of 14 v1.2 Exposure Draft
12. Table 2 Regression Statistics R Square 0.58 Standard Error 0.06 Observations 25.00 Durbin‐Watson 1.80 Coefficients S.E. Intercept (d’) 0.0000 Slope (c’) 0.08 0.01 t Stat 5.72 P‐value 0.00 95% conf. Figure 9 Regression Plot Equation 11 0.35 0.30 Change in log(Market cap) d2P/dt2 0.25 (1.00) 0.20 y = 0.0831x R² = 0.4292 0.15 0.10 0.05 (0.50)  0.50 1.00 1.50 2.00 2.50 (0.05) (0.10) Change in Metcalfe Value (n2) dn/dt CONCLUSION A key application of our findings is the ability to evaluate data and marketplace news with the intent to separate meaningful information from misleading noise. Noise is a dominant driver of price and Bitcoin Spreads Like a Virus March 20, 2019 Page 12 of 14 v1.2 Exposure Draft
13. volatility in the shortterm. It is not noteworthy that markets disseminate and consume noise about bitcoin or any other cryptocurrency. Forecasts of skyhigh prices and doomsday crashes are common with bitcoin. In many cases, cryptocurrency information (positive and negative) is crafted in a convincing way by experienced and knowledgeable sources and presented by reputable media. In isolation, these predictions are indistinguishable from information, even though they are probably noise. Though value is not observable, even an imperfect assessment of value serves to keep markets efficient. Over time price tends toward value. The model we have presented serves as a backdrop against which potential information can be evaluated. It does not predict that bitcoin’s price will soar or crash. Rather, it suggests that the probability of those extreme those events is very small because ultimately number of users drives price. To date, the typical approach to cryptocurrency valuation has been via Metcalfe’s law. Commonly expressed in shorthand as n2, it is the approximate value of P when n is large. We show that price is a function of n users, as Metcalfe’s law states. Our research differs from past models in that we derive that n may grow at a nonconstant rate over time, as a Gompertz function would indicate. This function, usually used to describe the growth of biological organisms like bacteria, tumors, and viruses, likely has some application to network economics, including cryptocurrency valuation. Lastly, we confirm past research that the longterm growth rate in users has considerable effect on the longterm price of bitcoin. REFERENCES Andolfatto, D., and A. Spewak. (2019). “Whither the Price of Bitcoin?” Economic Synopses. Federal Reserve Bank of St. Louis. 2019, No. 1. Bewley, R., and D. G. Fiebig. (1988). “A Flexible Logistic Growth Model with Applications in Telecommunications.” International Journal of Forecasting, Vol. 4, No. 2, pp. 177–192. Black, F. (1986) “Noise.” The Journal of Finance, Vol. 41, No. 3, pp. 528–543. Briscoe, B., A. Odlyzko, and B. Tilly. (July 2006). “Metcalfe’s Law is Wrong.” IEEE Spectrum, Vol. 43, No. 7 pp. 3439. Brunton, G. F., and T. E. Wheldon. (1980). “The Gompertz Equation and the Construction of Tumour Growth Curves.” Cell Proliferation, Vol. 13, No. 4, pp. 455–460. Chaddha, R. L., and S. S. Chitgopekar. (1971). “A ‘Generalization’ of the Logistic Curves and LongRange Forecasts (19661991) of Residence Telephones.” The Bell Journal of Economics and Management Science, Vol. 2, No. 2, p. 542 Gandal, N., J. Hamrick, T. Moore, and T. Oberman. (May 2018). “Price Manipulation in the Bitcoin Ecosystem.” Journal of Monetary Economics, Vol. 95, pp. 8696. Griffin, J.M. and A. Shams, “Is Bitcoin Really UnTethered?” (June 13, 2018). Available at SSRN: https://ssrn.com/abstract=3195066 or http://dx.doi.org/10.2139/ssrn.3195066 Islam, T, Fiebig, D.G., and Meade, N. (2002). "Modelling Multinational Telecommunications Demand with Limited Data." International Journal of Forecasting, Vol. 18, No. 4, pp. 605–624. Leising, M. (30 January 2018, 11:52AM). “U.S. Regulators Subpoena Crypto Exchange Bitfinex, Tether.” Bloomberg, www.bloomberg.com/news/articles/20180130/cryptoexchangebitfinextethersaidtogetsubpoenaedbycftc. Metcalfe, R. (2013) “Metcalfe's Law after 40 Years of Ethernet.” Computer, Vol. 46, No. 12, pp. 26–31. Murphy, J.S. (1973). “The Death Rate of Lysogenic Bacteria after Ultraviolet Induction Follows Gompertz's Law.” Mechanisms of Ageing and Development, Vol. 2, pp. 125–139. Pele, D.T. and M. MazurencuMarinescuPele. (2019). “Metcalfe's Law and Herding Behaviour in The Cryptocurrencies Market.” Economics Discussion Papers, No. 201916, Kiel Institute for the World Economy. http://www.economicsejournal.org/economics/discussionpapers/201916 Bitcoin Spreads Like a Virus March 20, 2019 Page 13 of 14 v1.2 Exposure Draft
14. Peterson, T. “Metcalfe's Law as a Model for Bitcoin's Value.” (2018). Alternative Investment Analyst Review, Q2, Vol. 7, No. 2, pp. 918. Robinson, M. and T. Schoenberg. (20 November 2018, 3:00AM) “BitcoinRigging Criminal Probe Focused on Tie to Tether.” Bloomberg, https://www.bloomberg.com/news/articles/20181120/bitcoinriggingcriminalprobeissaidtofocusontietotether. Scott, G.R. (1959) “Mortality of Rabbits Inoculated with Lapinised Rinderpest Virus.” Journal of Comparative Pathology and Therapeutics, Vol. 69, pp. 148–151. Van Hove, L. (July 2014). “Metcalfe's Law: Not so Wrong after All.” Netnomics, Vol. 15, No. 1. −−. (January 2016). “Metcalfe's Law and Network Quality: An Extension of Zhang et al.” Journal of Computer Science and Technology, Vol. 31, No. 1, pp. 117123. Zhang, X., J.L., and Z. Xu. (2015). “Tencent and Facebook Data Validate Metcalfe’s Law.” Journal of Computer Science and Technology, Vol. 30, No. 2, pp. 246–251. Bitcoin Spreads Like a Virus March 20, 2019 Page 14 of 14 v1.2 Exposure Draft
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